Approximately bisectrix-orthogonality preserving mappings

Ali Zamani

Commentationes Mathematicae, Tome 54 (2014) no. 2, pp. 167-176.

Résumé

Regarding the geometry of a real normed space \({\mathcal X}\), we mainly introduce a notion of approximate bisectrix-orthogonality on vectors \(x, y \in {\mathcal X}\) as follows:$${x\sideset{ ^{\varepsilon\!\!}}{}\perp}_W y \mbox{~if and only if~} \sqrt{2}\frac{1-\varepsilon}{1+\varepsilon}\|x\|\,\|y\|\leq \Big\|\,\|y\|x+\|x\|y\,\Big\|\leq\sqrt{2}\frac{1+\varepsilon}{1-\varepsilon}\|x\|\,\|y\|.$$ We study the class of linear mappings preserving the approximately bisectrix-orthogonality \({\sideset{ ^{\varepsilon\!\!}}{}\perp}_W\). In particular, we show that if \(T: {\mathcal X}\to {\mathcal Y}\) is an approximate linear similarity, then $${x\sideset{ ^{\delta\!\!}}{}\perp}_W y\Longrightarrow {Tx \sideset{ ^{\theta\!\!}}{}\perp}_W Ty \qquad (x, y\in {\mathcal X})$$ for any \(\delta\in[0, 1)\) and certain \(\theta\geq 0\).

Détail

DOI : 10.14708/cm.v54i2.699

Mots-clés : bisectrix-orthogonality, approximate orthogonality, isometry, orthogonality preserving mapping